U.S. Pat. No. 4,530,452, dated July 23, 1985, to Marik Balyasny and William F. Lovell, describes an apparatus for cleaving an optical fiber, which apparatus is also known as a cleaving tool. (The Marik Balyasny of said U.S. Pat. No. 4,530,452 and the Marik Balyasny of the instant patent application are one and the same person.) The instant invention described in this application performs the cleavage of the Balyasny-Lovell apparatus in an improved and novel manner.